To prove our results we follow the approach introduced in [1]}. Using concentration compactness techniques, we carry out a careful analysis of the behavior of minimizing sequences for the system (REF ) that have a specific type of symmetries, described in Section . The symmetries can be chosen to produce a change of sign by construction. As \(\varepsilon \rightarrow 0\)  the components of the system concentrate at fixed points of the group action. If the space of fixed points is trivial, all components will necessarily concentrate at the origin, but they will move far away from each other when the fixed-point space has positive dimension. A detailed description is given in Theorem REF .
